A plane flies 430 miles at a bearing of S27°E, then turns and flies N63°E for 135 miles. Find the distance and the bearing from the starting?
After you make your sketch, and after you enter your angles, you should realize that the plane makes a 90 degree turn.
So you just have a right-angled triangle.
Use Pythagoras to find the distance.
Let the angle opposite the 135 side be θ
then tanθ = 135/430
θ = ...
so your direction is S (θ+27) E
let me know what you got
Your distance of 450.69 miles is correct
However, the angle of 17.43 is the angle within the triangle.
You must add the original direction of 27 degrees to get S 44.3 E
To find the distance and bearing from the starting point, we can break down the given information into two components: the distance traveled in each direction, and the change in bearing.
First, let's calculate the distance traveled in each direction:
1. The plane flies 430 miles at a bearing of S27°E.
From the given bearing, we can determine that the plane is traveling south and slightly east.
To calculate the southward distance traveled, we can use trigonometry. The southward distance can be calculated as the product of the given distance (430 miles) and the cosine of the angle (S27°).
Southward distance = 430 miles * cos(27°)
Southward distance ≈ 387.04 miles
To calculate the eastward distance traveled, we can also use trigonometry. The eastward distance can be calculated as the product of the given distance (430 miles) and the sine of the angle (S27°).
Eastward distance = 430 miles * sin(27°)
Eastward distance ≈ 198.90 miles
2. The plane then turns and flies N63°E for 135 miles.
From the given bearing, we can determine that the plane is traveling north and slightly east.
To calculate the northward distance traveled, we can use trigonometry. The northward distance can be calculated as the product of the given distance (135 miles) and the cosine of the angle (N63°).
Northward distance = 135 miles * cos(63°)
Northward distance ≈ 58.03 miles
To calculate the eastward distance traveled, we can also use trigonometry. The eastward distance can be calculated as the product of the given distance (135 miles) and the sine of the angle (N63°).
Eastward distance = 135 miles * sin(63°)
Eastward distance ≈ 123.94 miles
Now, let's find the total distance and bearing from the starting point:
To find the total distance, we can add up the distances traveled in each direction:
Total distance = Southward distance + Northward distance
Total distance = 387.04 miles + 58.03 miles
Total distance ≈ 445.07 miles
To find the bearing from the starting point, we can use trigonometry. The bearing can be calculated as the arctangent of the eastward distance divided by the southward distance.
Bearing = arctan(Eastward distance / Southward distance)
Bearing = arctan(198.90 miles / 387.04 miles)
Bearing ≈ 26.65°
Therefore, the distance from the starting point is approximately 445.07 miles, and the bearing is approximately 26.65°.